interpoloid

Interpoloid is a geometric construct used in interpolation and modeling that generalizes the notion of a polyhedron by coupling a fixed set of vertices with a parametric family of weights. Given a finite vertex set V = {v1, ..., vk} in Euclidean space R^d and a parameter domain D in R^m, an interpoloid P is the set of points p(t) obtained as p(t) = sum_{i=1}^k w_i(t) v_i, where w_i(t) are polynomial basis functions defined on D and satisfy sum_i w_i(t) = 1 for all t in D and w_i(t) nonnegative where appropriate. As t ranges over D, the map t -> p(t) traces a family of convex combinations of the vertices, yielding a patch-like region that interpolates data assigned to the vertices.

If w_i are chosen as Bernstein polynomials on a simplex in D, the corresponding interpoloid reduces to a Bezier-like surface patch; if D is a simplex of dimension m, the standard simplex yields a Bezier simplex. More general choices of w_i produce other patches, including nonuniform and adaptive interpolations.

Properties of an interpoloid include that it is typically a convex set when coefficients are nonnegative and sum to one; the boundary is piecewise smooth. The construction can be extended to higher orders and to include weights that encode data values, yielding an interpolatory surface or volume.

Applications include computer graphics, geometric modeling, computer-aided design, finite element mesh generation, and data fitting. The concept relates to polytopes, barycentric coordinates, Bezier and NURBS constructions and serves as a flexible framework for interpolating values across a vertex-defined domain.